January 26

9.1 a) Find a number
$0\leq a < 73$
with
$a\equiv 9^{794} \mod 73$
(Hint: use Fermat's little theorem).

9.2 b) Show that for a prime number p the number (p-1)!+1 is divisible by p
(Hint: consider the number (p-1)! mod p. In this product one can split most of the residues to pairs of a residue and its inverse).

9.4 a) Assuming that the congruence
$7^{1734250}\equiv 1660565 \mod 1734251$
holds, show that 1734251 is a composite (i.e. non-prime) number.

c) The congruence
$2^{52632}\equiv 1 \mod 52633$
is true. Can you conclude that 52633 is a prime number?


10.3 Show that for
$m=561=3\cdot 11\cdot 17$
the congruence
$a^{m-1}\equiv 1 \mod m$
holds for every a relatively prime to m.
(Hint: use Fermat's little theorem to compute a^{m-1} mod p for every prime divisor p of m).

11.1 b) Determine the value of
$\phi(8800)$

11.5 Find x that solves
$x\equiv 3 \mod 37$
and
$x\equiv 1 \mod 87$